Macroscopic Interpretations of Microscopic Traffic Waves with Sparse Control

By Nour Khoudari

Traffic flow models help researchers understand vehicle movement on roads and forecast traffic states. Multiple varieties of these models exist at both microscopic and macroscopic scales. Vehicle-based microscopic models use ordinary differential equations (ODEs) to track the behavior of each individual vehicle and its interactions with other vehicles. In contrast, macroscopic models consist of partial differential equations that describe the evolution of aggregate quantities of traffic flow like density \(\rho\), flow rate \(q\), and bulk traffic velocity \(u\). Because many applications employ only one of these model types, we seek to contribute to the interplay between both perspectives.

Specifically, we can employ sampling and kernel density estimations to connect these two scales. We begin by deriving macroscopic quantities from microscopic vehicle trajectories. Given \(N\) vehicles at positions \(x_j(t)\) for \(j=1,\dots,N\) that travel at velocities \(\dot{x}_j(t)\), we can construct macroscopic field quantities via the kernel density estimate:

\[\rho(x,t) = \sum_{j=1}^N G(x-x_j),\quad q(x,t) = \sum_{j=1}^N \dot{x}_j G(x-x_j), \quad \text{and}\quad u(x,t) = q(x,t)/\rho(x,t).\]

We used the Gaussian kernel—defined as \(G(x) = (\sqrt{\pi}h)^{-\frac{1}{2}} \exp(-(x/h)^2)\)—with a kernel width \(h\) that is larger than the typical spacing between vehicles but smaller than the macroscopic length scales of interest. We specifically choose this method because the reconstructed fields exactly satisfy the continuity equation \(\rho_t+q_x=0\). 

To demonstrate the importance of satisfying the continuity equation for a traveling wave solution, consider a single wave profile that moves with speed \(s\) along the road; i.e., the quantities \(\rho\) and \(q\) only depend on a single variable \(\eta = x-st\). Then \(\rho_t = -s \frac{\mathrm{d}\rho}{\mathrm{d}\eta}\) and \(q_x = \frac{\mathrm{d}q}{\mathrm{d}\eta}\), and the continuity equation thus becomes \(\frac{\mathrm{d}}{\mathrm{d}\eta} (-s\rho+q) = 0\). Integration yields \(-s\rho+q = m\), where the integration constant \(m\) is the mass flux of vehicles relative to the wave. The resulting relationship \(q = m+s\rho\) means that pairs of density and flow rates \((\rho,q)\)—which are traveling waves that satisfy the continuity equation—form a straight line with a slope \(s\) that equals the speed of the traveling wave in the fundamental diagram space [2]. The fundamental diagram is a key tool in traffic research that describes the relationship between density and flow rate; an equilibrium solution on the fundamental diagram curve implies that the vehicles are equispaced and move at the same speed. However, equilibrium does not imply stability. Instead, certain model parameters determine the stability of equilibrium solutions.

Figure 1. Pairs of macroscopic quantities \((\rho, q)\) in the fundamental diagram plane. We used the intelligent driver model [4]—with a kernel width of \(h=20\) meters for a simulation of 60 vehicles on a 1,500-meter ring road that was initiated at equilibrium—to construct the plot. Figure courtesy of the author.

Figure 1 serves as an example of the reconstructed \((\rho,q)\) pairs for a microscopic simulation with a traveling wave solution; the blue dot denotes the pairs’ effective (average) state in the fundamental diagram. We begin with a microscopic simulation that has an unstable equilibrium initial state (red dot on the fundamental diagram curve in Figure 1) and seek to determine how the development of waves affects the average state (blue dot in Figure 1).

Our main goal is to better understand microscopic traffic waves and connect them to meaningful macroscopic interpretations. While some researchers utilize complex microsimulations to capture real traffic patterns, these methods are often unable to correctly attribute the reasoning behind such patterns. Therefore, we instead rely on simple single-lane simulations that isolate car-following behavior and fix certain quantities (density, flow rate, and velocity) to study their effects on the evolution of the effective state. 

When starting from equilibrium and adding perturbations to the system, we found that waves develop in the simulation and the effective state moves in the fundamental diagram space away from the initial equilibrium state and on a line that depends on the simulation’s initial setup. For simplicity, here we discuss the results for a ring road setup [3, 5]. In our single-lane simulations, vehicles are initialized at equilibrium in the unstable regime; the ring road length is fixed, and the last vehicle follows the first. To mimic real driving behavior, we introduce perturbations to the system by adding some scaled noise to every car’s velocity at each time step (solution of a stochastic ODE). This noise triggers instabilities and eventually causes the development of waves.

Figure 2. Trajectories of a simulation of 130 vehicles on a 1,500-meter intelligent driver model ring road. Noise is added for the first 100 seconds of the simulation to trigger instabilities. Figure courtesy of the author.

Figure 2 depicts wave development in vehicle trajectories on a ring road. By tracking the effective state for a ring road simulation of fixed density on the fundamental diagram level, we find that the effective state moves away from the initial equilibrium state and downwards on a vertical line as we add perturbations to the system and waves develop. This finding verifies the fact that the developing waves reduce the throughput on the road even though the density remains fixed.

An effective framework that helps interpret traffic waves can enable subsequent studies on other interesting situations. One relevant active area of research is the introduction of autonomous vehicles (AVs) on real roads or sparse controllers in simulations [1]. Since AVs will be on roadways in the future—and are already, if we consider existing consumer market vehicles with adaptive cruise control systems—exploring their usage for broader benefits, such as smoothing traffic flow, presents an intriguing alternative to costly traffic control infrastructure like ramp meters or variable speed limits. We can integrate a simple controller into our simulations by considering vehicles that act as AVs with an acceleration function \(\ddot x_{\text{AV}}\) that resembles the human driver acceleration \(\ddot x_{\text{human}}\) but with an additional relaxation term

\[\ddot x_{\text{AV}} = \ddot x_{\text{human}} + \gamma (U-\dot x),\]

where \(\gamma\) is the control gain and \(U\) is the AV’s desired velocity. 

The simulation in Figure 3 varies the choices of AV penetration rate and control gain \(\gamma\) for a fixed choice of desired velocity \(U\). A metric of success for the controller is the difference between the minimum and maximum velocity across all vehicles averaged over a final time interval. If that difference is close to zero, the controller has successfully dampened the waves and brought the system back to equilibrium; otherwise, it has failed. Integrating this simple controller into our simulations demonstrates that with the right parameter choice, we can dampen waves and sometimes even bring the system back to equilibrium with as little as a five percent AV penetration rate. However, there are practical limits on control gain values; for instance, ones that are too large result in nonphysical acceleration values [1]. On the level of the fundamental diagram, this limit implies that the effective state moves on the same line back to the initial equilibrium state as the controllers are activated and the waves are dampened.

Figure 3. Parameter analysis of autonomous vehicle penetration rate and control gain \(\gamma\) for fixed \(U\), conducted via a metric that measures the difference between the minimum and maximum velocities across all vehicles averaged over a final time interval. Figure courtesy of the author.

Our study utilizes kernel density estimates to derive macroscopic quantities from microscopic vehicle trajectories. This technique helps us analyze the macroscopic interpretations of microscopic waves and the effects of sparse control at the fundamental diagram level. Moving forward, we can potentially consider other aspects when connecting different scales of traffic models, such as the macroscopic limits of microscopic models in the unstable regime.

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References
[1] Cui, S., Seibold, B., Stern, R., & Work, D.B. (2017). Stabilizing traffic flow via a single autonomous vehicle: Possibilities and limitations. In 2017 IEEE Intelligent Vehicles Symposium (IV) (pp. 1336-1341). Redondo Beach, CA: Institute of Electrical and Electronics Engineers.
[2] Khoudari, N., & Seibold, B. (2022). Multiscale properties of traffic flow: The macroscopic impact of traffic waves. In Mathematics online first collections (pp. 1-30). Cham, Switzerland: Springer.
[3] Sugiyama, Y., Fukui, M., Kikuchi, M., Hasebe, K., Nakayama, A., Nishinari, K., … Yukawa, S. (2008). Traffic jams without bottlenecks — experimental evidence for the physical mechanism of the formation of a jam. New J. Phys., 10(3), 033001.
[4] Treiber, M., Hennecke, A., & Helbing, D. (2000). Congested traffic states in empirical observations and microscopic simulations. Phys. Rev. E, 62(2), 1805-1824.
[5] Wu, F., Stern, R.E., Cui, S., Delle Monache, M.L., Bhadani, R., Bunting, M., … Work, D.B. (2019). Tracking vehicle trajectories and fuel rates in phantom traffic jams: Methodology and data. Transp. Res. Part C: Emerg. Technol., 99, 82-109.

Nour Khoudari is a Ph.D. candidate in applied mathematics at Temple University. Her research focuses on traffic and energy modeling; she develops methodologies that connect macroscopic and microscopic traffic models to better understand nonlinear waves and builds simple but accurate fuel consumption models that are easy to integrate into optimization and control problems. Khoudari is a student researcher in the Congestion Impacts Reduction via CAV-in-the-loop Lagrangian Energy Smoothing (CIRCLES) project: a multi-institution endeavor that is funded by the Department of Energy and seeks to reduce traffic jams and fuel consumption through the introduction of autonomous vehicles.